ECON 416/516 - Elias
Lab 5 - Advanced Time Series Topics
115 points total
1. Use the monthly data in “wageprc.RData” for this question. For the U.S. economy,
let gpricet denote the monthly growth in the overall price level and let gwaget be the
monthly growth in hourly wages. These are both obtained as differences of logarithms:
gpricet = ∆ ln(pricet) and gwaget = ∆ ln(gwaget).
(a) (20 points) Estimate by OLS the following model:
gpricet = α0 + δ0gwaget + δ1gwaget−1 + δ2gwaget−2 + δ3gwaget−3 + δ4gwaget−4
+ δ5gwaget−5 + δ6gwaget−6 + δ7gwaget−7 + δ8gwaget−8 + δ9gwaget−9
+ δ10gwaget−10 + δ11gwaget−11 + δ12gwaget−12
which is a finite distributed lag with 12 lags of gwaget (the lags are located in the
data set). What are the estimated impact propensity and the long-run propensity?
Plot the estimated lag distribution (i.e., make a plot with the lag length on the
horizontal axis and the appropriate δˆi on the vertical axis) (Hint: to create the lag
length variable for the X-axis of the plot, use the R function “seq”).
(b) (20 points) Now estimate a simple geometric distributed lag model of gpricet on
gwaget. That is, estimate the following equation by OLS:
gpricet = α0 + γgwaget + ρgpricet−1 + νt
What are the estimated impact propensity and long-run propensity? Plot the esti-
mated lag distribution for periods zero through period 12 (i.e., 12 periods).
(c) (10 points) Compare the estimated IPs and LRPs of the finite distributed lag model
and the geometric distributed lag model. How do the estimated lag distributions
compare?
2. Use the yearly data on per capita investment in “hseinv.RData” for this question.
(a) (10 points) Test for a unit root in ln(invpct), where invpct is real investment per
capita. Include a linear time trend and two lags of ∆ ln(invpct). Use a 5% signifi-
cance level.
(b) (10 points) Use the approach from part (a) to test for a unit root in ln(pricet),
where pricet is the value of the housing price index.
(c) (5 points) Given the outcomes in parts (a) and (b), does it make sense to test for
cointegration between ln(invpct) and ln(pricet)?
3. (25 points) In class we conducted an Engle-Granger test to test for cointegration between
the general fertility rate (gfrt) and the real value of the personal tax exemption (pet) in
the U.S. Specifically, we ran the following regression,
gfrt = αt+ βpet + ut (1)
and conducted an augmented Dickey-Fuller test with no time trend and one lag on the
residuals.
Using the data in “fertil3.RData”, conduct an Engle-Granger test by performing the
following steps:
(i) Add t2 to equation (1), estimate the model by OLS, and obtain the residuals
(ii) Conduct an augmented Dickey-Fuller test with no time trend and one lag on the
residuals obtained in step (i).
The 5% critical value for the test is -4.15. What do you conclude?
4. (15 points) Use the quarterly data in “intqrt.RData” for this question. In class, we
estimated an error correction model for the holding yield on six-month T-bills (hy6t),
where one lag of the holding yield on three-month T-bills (hy3t−1) is the explanatory
variable. We assumed that the cointegration parameter was one in the equation hy6t =
α + βhy3t−1 + ut, and estimated the following error correction model:
∆hy6t = α0 + γ0∆hy3t−1 + δ(hy6t−1 − hy3t−2) + ut (2)
where ut has zero mean, given all hy3 and hy6 dated at time t − 1 and earlier. Now,
add to the model in equation (2) ∆hy3t−2 and (hy6t−2 − hy3t−3); that is, estimate the
model:
∆hy6t = α0 + γ0∆hy3t−1 + γ1∆hy3t−2 + δ(hy6t−1 − hy3t−2) + δ2(hy6t−2 − hy3t−3) + ut
(3)
To do this in R, create the variables “s 1” and “chy3 2”, where s 1 = lag(hy6,-2) -
lag(hy3,-3), and chy3 2 = lag(chy3,-2).
Are the coefficients on ∆hy3t−2 and (hy6t−2−hy3t−3) jointly significant at the 5% level?
What do you conclude about the appropriate error correction model? (Hint: Use the R
function “linearHypothesis” to conduct the F-test.)
Page 2